A capacitance of \(2\ μF\) is required in an electrical circuit across a potential difference of \(1.0\ kV\). A large number of \(1\ μF\) capacitors are available which can withstand a potential difference of not more than \(300\ V\). The minimum number of capacitors required to achieve this is

An automobile, travelling at \(40\, km/h\), can be stopped at a distance of \(40\, m\) by applying brakes. If the same automobile is travelling at \(80\, km/h\), the minimum stopping distance, in metres, is (assume no skidding)..........\(m\)

Two coherent sources of sound, \(S_{1}\) and \(S_{2}\) produce sound waves of the same wavelength, \(\lambda=1\, m ,\) in phase. \(S_{1}\) and \(S_{2}\) are placed \(1.5\, m\) apart (see fig.) A listener, located at \(L,\) directly in front of \(S_{2}\) finds that the intensity is at a minimum when he is \(2\,m\) away from \(S _{2}\). The listener moves away from \(S _{1},\) keeping his distance from \(S_{2}\) fixed. The adjacent maximum of intensity is observed when the listener is at a distance \(d\) from \(S\)\(_{1}\). Then, \(d\) is\(......m\)

Two long parallel conductors \(S_{1}\) and \(S_{2}\) are separated by a distance \(10 \,cm\) and carrying currents of \(4\, A\) and \(2 \,A\) respectively. The conductors are placed along \(x\)-axis in \(X - Y\) plane. There is a point \(P\) located between the conductors (as shown in figure). A charge particle of \(3 \pi\) coulomb is passing through the point \(P\) with velocity \(\overrightarrow{ v }=(2 \hat{ i }+3 \hat{ j }) \,m / s\); where \(\hat{i}\) and \(\hat{j} \quad\) represents unit vector along \(x\) and \(y\) axis respectively. The force acting on the charge particle is \(4 \pi \times 10^{-5}(-x \hat{i}+2 \hat{j}) \,N\). The value of \(x\) is

A drop of liquid of density \(\rho\) is floating half immersed in a liquid of density \(\sigma\) and surface tension \(7.5 \times 10^{-4}\,N\,cm ^{-1}\). The radius of drop in \(cm\) will be : (Take : \(g =10\,m / s ^{2}\) )

A \(0.5\) kg mass is in contact against the inner wall of a cylindrical drum of radius \(4\) m rotating about its vertical axis. The minimum rotational speed of the drum to enable the mass to remain stuck to the wall (without falling) is \(5\) rad/s. The coefficient of friction between the drum's inner wall surface and mass is _______. (Take \(g = 10\) m/s\(^2\))

The equation \(\lambda=\frac{1.227}{ x } nm\) can be used to find the de-Brogli wavelength of an electron. In this equation \(x\) stands for : Where,\(m =\) mass of electron \(P =\) momentum of electron \(K =\) Kinetic energy of electron \(V =\) Accelerating potential in volts for electron

Let \(f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}\) Then \(f\left(\frac{7 \pi}{12}\right) f "\left(\frac{7 \pi}{12}\right)\) is equal to

Let \(x =\sin \left(2 \tan ^{-1} \alpha\right)\) and \(y =\sin \left(\frac{1}{2} \tan ^{-1} \frac{4}{3}\right)\). If \(S =\left\{\alpha \in R : y ^{2}=1- x \right\}\), then \(\sum_{\alpha \in S } 16 \alpha^{3}\) is equal to \(...........\)

The least positive integral value of \(\alpha\), for which the angle between the vectors \(\alpha \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \mathrm{k}\) and \(\alpha \hat{\mathrm{i}}+2 \alpha \hat{\mathrm{j}}-2 \mathrm{k}\) is acute, is

If \(\sum_{r=1}^n T_r=\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)}{64}\), then \(\lim _{n \rightarrow \infty} \sum_{r=1}^n\left(\frac{1}{T_r}\right)\) is equal to :

Suppose \(f ( x )\) is a polynomial of degree four, having critical points at \(-1,0,1\) . If \(T =\{ x \in R \mid f ( x )= f (0)\},\) then the sum of squares of all the elements of \(T\) is

The wavelength of the photon emitted by a hydrogen atom when an electron makes a  transition from \(n =2\) to \(n =1\) state is ...... \(nm.\)